How do you find the exact value of cos(u-v) given that sinu=5/13 and cosv=-3/5?

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marfre Share
Mar 20, 2017

$\cos \left(u - v\right) = - \frac{16}{65}$

Explanation:

Find $\cos u$ using Pythagorean Theorem to find the adjacent side. Assume angle $u$ is in the first quadrant:

1. ${13}^{2} = {5}^{2} + {a}^{2}$
2. ${a}^{2} = 169 - 25 = 144$
3. $a = 12$
So $\cos u = \frac{12}{13}$

Find $\sin v$ using Pythagorean Theorem to find the adjacent side. Assume angle $v$ is in the second quadrant:

1. ${\left(- 3\right)}^{2} + {b}^{2} = {5}^{2}$
2. ${b}^{2} = 25 - 9 = 16$
3. $b = 4$
So $\sin v = \frac{4}{5}$

Use the difference formula $\cos \left(u - v\right) = \cos u \cos v + \sin u \sin v$:
$\cos \left(u - v\right) = \frac{12}{13} \cdot - \frac{3}{5} + \frac{5}{13} \cdot \frac{4}{5} = - \frac{36}{65} + \frac{20}{65} = - \frac{16}{65}$

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